Tricky Fourier Transform problem for an exponential function

In summary, the Fourier transform of the function f(t)=2 \pit^{2}e^{- \pi t^{2}} using the definition of the Fourier transform is (1-2{\pi \nu^{2}})e^{- \pi \nu^{2}}. To solve the integral, you can use integration by parts on the expression \int_{-\infty + i \nu}^{\infty + i \nu} 2 \pi x^2 e^{-\pi (x^2 + \nu^2)}\, dx .
  • #1
eaglemike
1
0

Homework Statement


find the Fourier transform, using the definition of the Fourier transform [itex]\widehat{f}[/itex]([itex]\nu[/itex])=∫[itex]^{∞}_{-∞}[/itex]f(t)e[itex]^{-2 \pi i \nu t}[/itex]dt, of the function f(t)=2 [itex]\pi[/itex]t[itex]^{2}[/itex]e[itex]^{- \pi t^{2}}[/itex]

Homework Equations



I have the answer:

(1-2[itex]{\pi \nu^{2}}[/itex])e[itex]^{- \pi \nu^{2}}[/itex]

The Attempt at a Solution



After inserting f(t) into the equation for the transform, I added the exponents on the e terms, factored out -[itex]\pi[/itex], and added and subtracted [itex]\nu^{2}[/itex] to get

∫[itex]^{∞}_{-∞}[/itex]2[itex]\pi[/itex]t[itex]^{2}[/itex]e[itex]^{- \pi (t^{2}+2 i \nu t + \nu^{2} - \nu^{2})}[/itex]dt

I then substituted x=t+i[itex]\nu[/itex] to get

∫[itex]^{∞}_{-∞}[/itex]2[itex]\pi[/itex](t+i[itex]\nu[/itex])[itex]^{2}[/itex]e[itex]^{- \pi (x^{2}+ \nu^{2} )}[/itex]dt

at this point I know that I should be able to solve the integral, probably by integrating by parts, but I am really just lost. This seems like such a tricky integral. I thought maybe squaring the equation to get a double integral and then converting to polar coordinates would work but I couldn't get that to work out either. Thanks for the help!
 
Last edited:
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  • #2
eaglemike said:

Homework Statement


find the Fourier transform, using the definition of the Fourier transform [itex]\widehat{f}[/itex]([itex]\nu[/itex])=∫[itex]^{∞}_{-∞}[/itex]f(t)e[itex]^{-2 \pi i \nu t}[/itex]dt, of the function f(t)=2 [itex]\pi[/itex]t[itex]^{2}[/itex]e[itex]^{- \pi t^{2}}[/itex]

Homework Equations



I have the answer:

(1-2[itex]{\pi \nu^{2}}[/itex])e[itex]^{- \pi \nu^{2}}[/itex]

The Attempt at a Solution



After inserting f(t) into the equation for the transform, I added the exponents on the e terms, factored out -[itex]\pi[/itex], and added and subtracted [itex]\nu^{2}[/itex] to get

∫[itex]^{∞}_{-∞}[/itex]2[itex]\pi[/itex]t[itex]^{2}[/itex]e[itex]^{- \pi (t^{2}+2 i \nu t + \nu^{2} - \nu^{2})}[/itex]dt

I then substituted x=t+i[itex]\nu[/itex] to get

∫[itex]^{∞}_{-∞}[/itex]2[itex]\pi[/itex](t+i[itex]\nu[/itex])[itex]^{2}[/itex]e[itex]^{- \pi (x^{2}+ \nu^{2} )}[/itex]dt

at this point I know that I should be able to solve the integral, probably by integrating by parts, but I am really just lost. This seems like such a tricky integral. I thought maybe squaring the equation to get a double integral and then converting to polar coordinates would work but I couldn't get that to work out either. Thanks for the help!

Your last expression is wrong: it should be [itex] \int_{-\infty + i \nu}^{\infty + i \nu} 2 \pi x^2 e^{-\pi (x^2 + \nu^2)}\, dx , [/itex], which can be replaced by the same integral from x = -infinity to +infinity along the real x-axis (why)? Now int x^2*exp(-x^2) dx can be attacked using integration by parts.

RGV
 
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1. What is the Fourier Transform and how does it relate to exponential functions?

The Fourier Transform is a mathematical operation that decomposes a function into its constituent frequencies. It is commonly used in signal processing and image analysis. The Fourier Transform of an exponential function is a complex function that consists of a peak at the frequency of the exponential and a series of harmonics at multiples of that frequency.

2. Why is the Fourier Transform of an exponential function considered a tricky problem?

The Fourier Transform of an exponential function involves complex numbers and can be difficult to calculate or interpret. It also requires knowledge of advanced mathematical concepts such as complex analysis and Fourier series, making it a challenging problem for many scientists.

3. What are some applications of the Fourier Transform for exponential functions?

The Fourier Transform of an exponential function is used in a variety of fields, including signal processing, image analysis, and quantum mechanics. It is also commonly used in electrical engineering to analyze the response of circuits and filters to exponential input signals.

4. How can I solve a tricky Fourier Transform problem for an exponential function?

To solve a tricky Fourier Transform problem for an exponential function, you will need to use complex analysis techniques and have a solid understanding of Fourier series. It is also helpful to have access to mathematical software or tools such as MATLAB to assist with calculations.

5. Are there any alternative methods for solving a Fourier Transform problem for an exponential function?

Yes, there are alternative methods for solving a Fourier Transform problem for an exponential function. One approach is to use the Laplace Transform, which is closely related to the Fourier Transform and can sometimes be simpler to calculate. Another option is to use numerical methods or approximations to get an estimate of the Fourier Transform.

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